JoSS Article: Volume 2

Abstract: We propose a novel visualization
approach that facilitates graphical exploration and communication
of relative actor status in social networks. The main idea is to
map, in a drawing of the entire network, actor status scores to
vertical coordinates. The resulting problem of determining
horizontal positions of actors and routing of connecting lines
such that the overall layout is readable is algorithmically
difficult, yet well-studied in the literature on graph drawing.
We outline a customized approach.

The advantages of our method are illustrated in a study of
policy making structures from the privatization processes of
former East German industrial conglomerates, in which the visual
approach led to additional findings that are unlikely to have
been revealed using non-visual means of analysis.

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From the very beginning, visualization has been an essential
tool in the analysis of social networks. In his groundbreaking
work, Moreno (1953) relied
extensively on graphical representations, and there is hardly any
mentioning of alternatives to visual analysis of sociometric
data. In fact, he attributes the breakthrough of the sociometric
movement to a showing of sociometric charts at the 1933
convention of the Medical Society of the State of New York (1953, p. xiii).

While early network analysis was largely based on plausible,
yet informal, concepts and qualitative data exploration, a wealth
of formal concepts has been subsequently developed to provide
quantitative empirical evidence for substantive research
questions. For a comprehensive overview of such methods see Scott (2000) or Wasserman and Faust (1994).
Sadly, visualization techniques have not kept up with this
progress in measurement, leading to a divergence of analysis and
graphical presentation that can be traced through the history of
social network visualization (Klovdahl,
1981; Brandes et al.,
1999; Freeman, 2000).

While today visualizations are used to present network data,
quantitative results of network analyses are still typically
given in tabular form. However, aggregate indices in general are
insufficient to fully appreciate and understand the structural
information contained in network data. In network analysis, it is
therefore desirable to integrate graphical presentation of the
actual network and results from quantitative analyses.

Many types of networks are traditionally visualized using
point-and-line representations (Bertin, 1983). Since few networks
have an underlying spatial layout, their elements need to be
positioned in some other meaningful way. While the tedious work
of manually positioning the elements is out of the question even
for small to medium-size networks, the primary design principle
implemented in currently available software for automatic layout
is clarity. That is, the focus is on readability rather than
visual communication of substantive content.

In addition to the inherent difficulty of laying out an
abstract network in a readable way (see Di Battista et al., 1999,
or Kaufmann and Wagner, 2001, for
overviews of algorithms for the visualization of networks in
general), there is also the issue of confidence. Who is going to
comfortably draw conclusions from complex aggregate data, if it
is difficult to relate them to the original network data and if
it is unclear how much the drawing of the network leads to wrong
impressions and succeedingly to wrong interpretations?

We argue that it is both useful and feasible to reintegrate
formal network analysis and graphical presentation. Our approach
is to contextualize analytic results with the underlying network
data by parameterizing the graphical design of visualizations
with structural properties. In other words, we want to `explain'
derived quantities by showing them simultaneously with the data
in a single diagram. To be effective, such diagrams must be
grounded on some express design principles. We thus follow
recommendations from Tufte (1997).

A simple example of explanatory visualization of attribute
data is the depiction of a mean as a horizontal line through a
bar chart of its constituent values. Since the mean as the
aggregate index can result from quite different data, showing the
data together with the index can be viewed as an explanation of
the latter.

Figure 1: Simultaneous display of data points
and their mean value

More sophisticated examples of visualization strategies that
could be called explanatory are mostly concerned with clustering.
The common design principle used to convey a semantic (given) or
syntactic (structural) clustering is spatial proximity supported
by delineation of cluster boundaries or shading of cluster
backgrounds (see, e.g., Frank 1996;
Krempel 2000; de Nooy et al. 2000).
The only explanatory visualization of exact actor indices that we
know of is the mapping of structural centrality to geometric
centrality introduced in Brandes
et al. (1999b). In the present paper, we propose a
corresponding method for explanatory visualization of status
indices.

By appropriate formalization and using advanced algorithms,
explanatory visualizations can be produced automatically, thus
shifting the production of information-dense, yet readable,
network graphics from the artistic to the scientific domain, the
implications being increased reliability and easy
reproducibility.

The benefits of automatically produced visualizations based on
substantive perspectives are two-fold: On the one hand, they
facilitate effective communication of findings, but on the other
hand they also facilitate graphical exploration of network data.
In this paper, our focus is on the second aspect. The argument is
illustrated by a study of policy making structures, in which the
visual approach led to additional findings that are unlikely to
have been revealed using non-visual means of analysis. One
in-depth rather than a series of superficial examples is chosen
to demonstrate how substantive research questions - in this case
about political decision processes - are connected to design
principles to form an explorative visualization tool for the
analysis.

The paper is organized as follows. In Section 2, we develop a graphical design
for explanatory visualization of networks and status therein. An
algorithm to produce such drawings is outlined in Section 3. In Section 4, we demonstrate the
advantages of graphical status exploration by presenting a study
of policy networks, in which the power structure of actors
involved in the privatization processes of the former East German
steel and shipbuilding industries is investigated. Several
conclusions are offered in Section 5.

The goal of this work is to provide automatic visual support
for systematic status exploration in social networks. To our
knowledge, until now such analysis has not been performed using
automatically produced diagrams, presumably because of a lack of
visualization principles that are general enough to facilitate
unbiased interpretation. There are three main aspects such
principles need to address (Brandes et al., 1999a):
the substantive content to be visualized, the graphical design,
and the algorithm realizing it. Substance and design are
described in this section, and a corresponding algorithm is
sketched in the next.

When exploring his hand-drawn sociograms, Moreno (1953) used a very simple
concept of status, sociometric choice, which would now be called
weighted indegree. Since its constituent factors are simply the
choices an actor receives from alters, sociometric choice is a
local measure and hence easily recognized in point-and-line
diagram. Therefore, no sophisticated design of a sociogram is
required to visualize choice status.

Figure 3:
Historical example of a status-ordered matrix with
high-status actors (class I) at the top and to the
left, and low-status actors (class V) at the bottom
and to the right. Instead of showing them individually,
the total number of choices in each block is given. Note
the tendency to choose actors with higher status (adapted
from Longmore 1948)

In Whyte (1943), status
is an extrinsic property of network actors, but Whyte
nevertheless integrates it in the design of several sociograms by
arranging actors vertically so that positions indicate relative
actor status. As can be seen in the example shown in Figure 2, status is communicated quite
effectively. It should be noted, however, that the network does
not form the basis for this hierarchy, since status is determined
by other factors. In this sense, the visual hierarchy and the
network are independent, but it is a conclusion that can be drawn
from the visualization that connections rarely exist between
actors of significantly different status.

Similarly, Longmore (1948)
introduces a semantic status attribute into matrix
representations by ordering actors along the diagonal according
to their rank, with the highest-ranking actors in the upper left
and lowest-ranking actors in the lower right corner. Status
classes thus correspond to blocks along the diagonal
(Figure 3). Loomis and Powell (1949)
adapt this design to sociograms in which classes correspond to
horizontal stripes, again with high-status classes shown above
low-status classes. Positions within a class are determined to
reduce visual noise caused by connecting lines.

The first and only attempt we know of to simultaneously
visualize structurally defined status and connections determining
it, is from Northway (1954).
She also uses horizontal stripes, but to indicate the quartile in
which the sociometric choice status of an actor lies. However,
only a focal actor and its alters are shown, probably because the
arrangement of an entire network in this fashion is too
cumbersome. Note that positioning of actors and routing of
connections had to be carried out by hand in all of the above
visualization procedures.

The first non-trivial formalization of structural status in a
network is attributed to Katz (1953),
who developed an index that takes into account not only direct
choices made by alters, but also indirect choices by other
actors. The result is a number assigned to each actor, indicating
relative status in the network. However, this is exactly where
formal analysis and visual exploration begin to diverge.
Incidentally, Katz is also a prominent figure arguing in favor of
representing networks by matrices rather than sociograms (1947).

Figure 5:
Formal organizational chart and adjacency matrix of
advice relationship. A matrix entry of 1 indicates
that the row actor turns to the column actor for advice

manager

Manuel

0

0

0

0

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0

0

0

0

1

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0

supervisors

Charles

1

0

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1

0

0

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0

1

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Donna

1

0

0

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0

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1

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Stuart

1

1

0

0

0

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0

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1

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auditors

Bob

0

0

0

1

0

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1

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1

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0

Carol

0

1

0

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0

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0

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Fred

0

0

0

1

0

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0

0

0

0

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Harold

0

1

0

0

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0

0

0

0

0

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Sharon

0

0

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1

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Wynn

0

1

0

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Kathy

0

0

1

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1

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1

secretaries

Nancy

0

0

1

0

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Susan

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1

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1

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1

Tanya

0

0

1

0

0

0

0

0

0

0

1

1

0

0

The difficulty of recognizing aggregate data in the absence of
diagrams that explicitly represent them is nicely illustrated in
a story from Krackhardt (1996),
which we now use as an example to motivate our own approach.
Krackhardt analyzed a group of 14 employees, the internal
auditing staff of a large company. The group's formal
organization is compared to an informal relation called `advice,'
i.e. who does an actor turn to for help or advice at work about
work-related questions or problems? Organizational and advice
relation data are given in Figure 5.

Commonly used network visualization tools such as Pajek (Batagelj and Marvar, 1998),
KrackPlot (Krackhardt
et al., 1994), or MultiNet (Richards and Seary, 1999) are
designed to produce general purpose visualizations focusing on
the ease of perceiving connectedness information (i.e. the
presence or absence of links between pairs of actors), or
inherent symmetry. Node positions are typically determined using
variants of the spring embedder (Eades, 1984),
multidimensional scaling, or eigenvectors of network-related
matrices such as the adjacency or Laplacian matrix. Pajek (Batagelj and Marvar, 1998) provides
the option to fix one or more dimensions of the layout space,
e.g. by mapping one or more node indices to coordinates, but
currently has no dedicated algorithm to produce readable
visualizations given such constraints. Since the result of the
status analysis cannot be taken into account with the common
layout algorithms, status indices need to be represented by the
size of nodes, by numerical labels, or separate from the drawing
as in Figure 6.

Though the network diagram is very readable, it does not
convey the interesting substantive information. Moreover, its
design is inherently undirected (the picture would be the same
even if some or all of the edge directions were reversed), and it
is next to impossible to relate the status scores to the picture.
Assume, for instance, we swap the status scores of Nancy and
Manuel; the visualization would not provide any indication that
something was wrong.

This is in stark contrast to empirical evidence suggesting
that network layout not only affects the ease of reading (Purchase et al., 1997),
but has an influence on the understanding and interpretation of
substantive content as well (McGrath
et al., 1997). Consequently, Krackhardt (1996) arranged
the actors so that most nominations point in an upward direction,
thus creating an informal advice hierarchy that yields an
implicit notion of status (see Figure 7).

The advice hierarchy largely resembles the formal
organizational hierarchy, with one notable exception. Confronted
with the graphical evidence, the manager concluded that changes
he introduced to increase through-put may have been ineffective
because he had not made sure that the secretary presiding the
informal hierarchy of advice was backing them (Krackhardt, 1996,
p. 166f).

Though it works fine in this particular example, note that the
above rule for vertical arrangement is error-prone in general,
since the requirement of a maximum number of upward oriented
connections may result in misleading visual explanations. A
simple example of this kind is a network of actors whose
connections form a directed cycle. Any one connection can be
chosen as the single downward oriented one, but each choice
results in a different vertical ordering of the actors. See the
paragraph on layer assignment in Section 3 for more details on this
problem.

Krackhardt's visualization is based on the same principle as
Whyte's in Figure 2 and
Northway's in Figure 4.
While using different concepts of status, they all refer to the
everyday notion of `higher' and `lower' status by mapping status
to vertical positions. These approaches have, however, two
limitations that need to be overcome: First, the mapping of
status to vertical positions is somewhat arbitrary, since there
are no guidelines on how to determine the exact vertical position
from a status classification or relative ordering; and, second,
horizontal positions serving to produce a readable drawing are
determined manually.

We overcome these limitations, and thereby reintegrate formal
analysis and graphical presentation, by placing actors at
vertical positions that exactly represent their status score, and
by determining horizontal positions algorithmically in such a way
that the overall visualization is readable.

The idea is illustrated in Figure 8, where actors are assigned y-coordinates
that represent exactly their Katz status score in the network.
Note how the vertical ordering differs from that in Figure 7. While the stem-and-leaf
diagram of Figure 6 does
indicate this fact as well, the visualization also explains the
reason why: the definition of Katz's status index implies that
Nancy's sole nomination of Donna results in Donna's status being
higher than Manuel's.

In principle, any definition of status, be it structural or
external, can be used in this approach, provided it translates
into numerical values specifying y-coordinates. We refer
the reader to Chapter 5 of Wasserman and Faust (1994)
for an overview of structural status concepts, but as already
pointed out for the criterion of upward pointing links, we
caution that not every definition leads to reliable explanations.

Some additional information is given in the
visualization of Figure 8
by depicting actors with ellipses rather than circles. This way,
the ratio of incoming and outgoing links is incorporated into the
drawing without changing the layout. Let in(a) and out(a)
denote the in- and outdegree, i.e. the number of incoming and
outgoing connections, of an actor a. Then, a horizontal
radius h(a) and a vertical radius v(a)
for the ellipse are chosen to satisfy so that the ratio of in- and outdegree
is visually represented by the ratio of height and width, and the
sum of the degrees is represented by the area of the node. A
minimum height and width is used for zero in- and/or outdegree,
and simple adjustments of the second equation account for node
shapes other than ellipses (rectangles, rhombs, etc.).

Other than substantive, there are ergonomic criteria
visualizations should satisfy. For example, a large number of
crossing lines makes a drawing difficult to read (Purchase et al., 1997).
Visualizations like the one in Figure 8 are therefore more difficult
to produce than, e.g., bar charts, because we can not just place
actors at the specified y-coordinates with some trivially
determined x-coordinates. An algorithm to generate
readable drawings under the substantive constraint of status
mapped to vertical positions is described in the next section.

To automatically generate layered visualizations of social
networks, it is not sufficient to require actors to lie on a
horizontal line with a given y-coordinate (representing
the actor's status). We also have to provide an algorithm to
compute x-coordinates for actors, and bend points for
connections in the network. This is a special case of a graph
drawing problem, where a graph is a collection of vertices or
nodes (here representing the actors) and edges or lines (here
representing the connections between actors). Di Battista et al. (1999)
and Kaufmann and Wagner (2001)
provide good overviews of the field.

The most commonly used framework for horizontally layered
drawings of graphs is presented in Sugiyama et al. (1981).
It consists of the following generic steps:

determine a layer for each node,

subdivide each connection by bend points at each layer it
crosses and determine, within each layer, the relative
ordering of nodes and bend points, and finally

assign x- and y-coordinates to each node or
bend point.

Steps 2 and 3 are separated to enable the use of
combinatorial methods in the second step, which serves to reduce
the number of crossing lines. Note that crossings severely affect
the readability of a drawing (Purchase, 1997), and that the
number of crossings between two adjacent layers is fully
determined by the relative ordering of nodes and bend points,
independent of the actual coordinates (hence the introduction of
bend points, see Figure 9).

There is a whole range of implementations, the most widely
used being the dag system (Gansner
et al., 1988). Since, however, we do not know of an
available system which applies the Sugiyama approach to layered
graphs with pre-specified vertical coordinates, we will outline a
customized variant. Comprehensive overviews of other approaches
to carry out the above steps are given in Chapter 9 of Di Battista et al. (1999)
and Bastert and
Matuszewski (2001).

Figure
9: A three-layer graph with many crossing
lines, and the same graph after subdivision of lines at
layers and reordering of nodes and bend points. Note that
bend points are not shown in the final visualization

Layer assignment. A fairly common approach to layering
is to break all directed cycles, if any, by temporarily reversing
some connections, and to assign nodes to layers by topological
sorting. Reversing the minimum number of connections nicely
corresponds to finding a layering with a maximum number of upward
pointing arcs.

Though intuitively an appealing idea and successfully employed
in Krackhardt's drawing of the advice network, it should not form
the basis of automatically generated status visualizations. There
are three substantive arguments against this approach if we are
to support exploratory data analysis.

Firstly, the implicit definition of status (directed lines
imply that the receiver has a higher status than the sender)
yields only a partial ordering, whereas y-coordinates
impose a complete ordering. Secondly, a minimum cardinality set
of cycle breaking connections need not be unique, thus only one
of potentially many equally valid interpretations is suggested.
Just consider a network consisting of a directed cycle only. And
thirdly, the problem of determining such a set with minimum
cardinality is NP-hard (Karp,
1972), i.e. likely to be computationally intractable.
The results of any heuristic or approximation algorithm suffer
from the same problem encountered for multiple optimal solutions,
namely a potential multitude of contradictory results, and an
uncertainty of whether the suggested interpretation is granted.

As an illustration, consider the layered drawing of the advice
network in Figure 10,
which was produced using a standard instance of the Sugiyama
framework with a heuristic layer assignment procedure implemented
in the AGD library of algorithms for graph drawing (Mutzel et al., 1998). By all
means, the resulting visualization is readable. Substantively,
however, it is grossly misleading, because it would have
suggested to Manuel that he is well in control of his auditing
group.

In summary, all three of the above aspects thus introduce
arbitrariness into the complete ordering of actor status that a
layering implies. Interpretation of relative status becomes
unreliable, if not impossible, in visualizations based on a
maximum number of upward pointing arcs, and only one notion of
status is supported.

Assuming that formal status indices have a sound theoretical
basis (a discussion of the appropriateness of an interval scale
measurement is beyond the scope of this paper), any such index
can be used for the y-coordinate of each node (subject to
scaling) and thus also to determine a layering. A trivial layer
assignment is to partition the nodes into sets of equal status,
and place each set in its own layer, vertically ordered with
respect to the status index. Status values often differ only
marginally, though, leading to very close layers that cause
perceptual problems like, e.g., several crossing (or
non-crossing?) line segments running almost horizontally (see
Figure 11). To
avoid such problems, status values are clustered and all vertices
with status values in the same cluster are assigned to the same
logical layer (without changing their y-coordinates).
Though any clustering may be used, the examples in this paper
were prepared with an agglomerative clustering scheme starting
with singletons and merging two layers, if the minimum status
difference between any pair of nodes in different layers is below
some threshold depending on the number of nodes in the network.

Crossing reduction. In this step, we are given a
layering of the nodes and introduce bend points where lines need
to cross a layer. Our goal is to find horizontal orderings of
nodes and bend points in each layer such that the number of
crossing lines is small. Note that the number of crossings
depends only on the ordering, not the actual coordinates.

Finding an ordering that minimizes the number of crossing
lines is another NP-hard problem (Garey and Johonson, 1983),
but this time it only affects readability rather than
interpretation. A common heuristic is the layer-by-layer sweep,
in which the ordering in, say, the first layer is fixed and the
second layer is reordered to reduce the number of crossings.
Then, the order in the second layer is fixed, and the third layer
is reordered, and so on. After reaching the last layer, the
process is reversed and repeated up and down the layering until
it does not yield further improvement. Note however, that even
minimizing the number of crossings between neighboring layers,
where the ordering in one layer is fixed, is NP-hard (1994).

In practice, the two-layer problem can be solved optimally for
medium-size instances using a computationally involved method (Jünger and Mutzel, 1997).
Since in general the overall number of crossing will not be
minimum anyway, we use the simpler barycenter heuristic, placing
a node at the average position of its neighbors in the next
layer, to obtain an initial ordering. Then, global sifting (Matuszewski et al.,
1999) is applied to further reduce the number of remaining
crossing. Roughly speaking, global sifting picks one node at a
time and finds the locally optimal position within a layer by
probing all of them. In combination, these heuristics perform
quite satisfactory, and our experiences suggest that the
additional effort caused by sifting is indeed worth it.

Horizontal placement. Given y-coordinates, a
layering, and an ordering of nodes and bend points within each
layer, it remains to compute actual x-coordinates
respecting the horizontal orderings. Pleasing visualizations are
obtained by ensuring that long lines run vertical as much as
possible, and reducing the horizontal distance spanned when it is
not. While for this paper we modified an implementation of a fast
heuristic from Buchheim
et al. (2001), we are now moving to a simplified
approach (Brandes and
Köpf, 2001).

In the following section the visualization approach we have
presented will be illustrated with an analysis of two policy
networks. It will be shown how explanatory visualization can not
only be used for communication of data and results but also for
the exploration of data that will enable researchers to discover
structural properties that could otherwise easily go unnoticed.
In order to clarify our arguments it is necessary to give a short
description of the research background of the study which
represents the substance for the visualization. We will then draw
some general conclusions about the visualization of status
indices.

The two policy networks presented are conceptualizations of
multi-actor systems of public, societal and private organizations
that developed during the privatization of two industrial
conglomerates in East Germany as part of the economic
transformation after German unification in 1990. Their
privatization is understood as political bargaining processes
between a multitude of actors that are connected by several
different kinds of ties.

At the heart of the economic transformation in East Germany
was a large-scale privatization process. This privatization was
foreseen to be carried out by the Treuhandanstalt (THA), a public
agency of the federal government. It was the owner of all assets
that belonged to the people's property (Volkseigenes
Vermögen) of the former German Democratic Republic (GDR).
Due to its institutional position and its ownership of all
companies, it was generally assumed to be one of the most
powerful actors in the transformation of East Germany.

The decision as to which investor could acquire which
enterprise under what conditions and the question which
enterprises should survive at all lay generally in the discretion
of the THA. This powerful position was presumably fundamentally
altered in the cases of `big politicized privatizations,' e.g.
privatizations which gained an enormous and sometimes nationwide
importance, because their outcomes affected the future of whole
regions. This was basically the case with the industrial
megaconglomerates (Kombinate), which regularly
represented the economic core in otherwise economically weak
regions encompassing tens of thousands of employees who were
often geographically highly concentrated.

Intensive and conflict ridden political bargaining processes
developed between a multitude of actors around the question what
to do with these `dinosaurs of socialist industrial policy' in
the future (big politicized privatizations). Among these actors
were political as well as private actors from the various
administrative levels within the European Union and Germany. The
questions what power structure would evolve and which actors
would primarily determine the outcome were especially
interesting, because the THA was the single dominant actor in the
mass privatization cases and these bargaining systems only
developed after 1990 encompassing `old' actors in the respective
industrial sectors from West Germany and `new' actors in East
Germany. The failure of these privatizations could have led to a
serious disruption of the whole transformation. They were
therefore seen as highly important political cases for the
overall transformation strategy and had a high level of priority
for the involved actors.

The central research questions of the study therefore were:

What kind of policy making structures evolved during the
decision processes, i.e. what actors would be involved,
what coalitions would emerge, etc.

What were the actors' power positions and the power
concentration in these networks?

These questions were studied in two case studies: the
privatization of the shipbuilding industry and of a major steel
plant (EKO Stahl AG). In the boundary specification process,
27 actors in the case of the shipbuilding industry and
21 actors in the case of the steel plant could be identified
who sought to directly or indirectly influence the decisions on
privatization and restructuring.

In both cases the type of actors were quite similar, sometimes
they were even identical in personae. Actors who were part of the
bargaining system were the European Commission (General
Directorate for Competition), the federal ministries of finance
and economics, the federal chancellery, the respective East
German state government, parties within the state parliament, the
board of directors and the supervisory board of the
Treuhandanstalt, the local governments with enterprise sites, the
board of directors, supervisory board and the workers' council of
the East German enterprise, the metal workers' union, competitors
in West Germany and West German state governments with
competitors' sites. Therefore, it can be stated that the original
governance structures of mass privatizations in which the
Treuhandanstalt was the single dominant actor were significantly
differentiated in regard to the number, type and functions of the
actors.

The policy making system was described and analyzed on the
basis of several types of ties based on communication, exchange
of resources, consideration of interest, etc. In the following,
however, we confine ourselves to the analysis of the power
structures resulting from `obligation of report' and
`consideration of interest' with the help of status
visualizations.5

The actors' structural status is used as an indicator for
power/influence in policy networks. The status index determined
for these networks is from Burt
(1982) and was computed using the software program STRUCTURE
(Burt, 1991). For unweighted
networks, this index assigns each actor a status score that is
the weighted average of the status scores of those actors
choosing it. The contribution of each chooser is weighted by the
inverse of its outdegree. This can be regarded as a very useful
operationalization of power or influence in political processes.
Power and influence of an actor in a decision process not only
differs according to how many other actors take her interests
into account but also how much the interests of third actors find
their way into a decision process as well. Therefore Burt's power
index implemented in STRUCTURE (Burt, 1991) was chosen because it
includes this understanding. But it should be re-emphasized here
that the visual analysis could be carried out with any index
which determines the rank prestige of actors according to some
substantive research question.

The structure based on the tie `consideration of interest' is
regarded as the final power structure in democratic political
decision processes, which is based on elements of influence
(centrality in communication networks) and domination/coercion (Knoke, 1990), which was
operationalized as the actors' status in the network based on
`obligation of report'. Actors who mandatorily receive reports
from others usually have the right to judge or give orders to
these other actors, thus indicating an asymmetric power position
based on coercive elements. In order to focus on the hierarchical
layout, `influence' based on centrality is not reported here.

The differentiation/concentration of power can be measured by
looking at the hierarchization of the decision system.
Unfortunately, unlike the generic network centralization index (Freeman, 1979), no generally
applicable network hierarchization index is available. For the
purpose of this research, hierarchization is measured using the
index suggested in Krackhardt
(1994). It is determined by subtracting the ratio of the
number of pairs of mutually reachable actors (i.e. those pairs
connected by directed paths in either direction), and the number
of pairs of connected actors (i.e. those pairs connected by any
sequence of links) from one. If all connected pairs are mutually
reachable, the level of hierarchization equals zero. If no such
pairs exist in a structural constellation, the index equals one
and the network forms a perfect hierarchy.

Status visualizations are given in Figures 12 and 13. To reduce clutter due to
bidirectional edges and arrow heads, non-downward pointing
uni-directional edges are depicted in black, bidirectional edges
in green, and downward pointing edges in red. Thus, the existence
and direction of a choice is indicated. As application-specific
information, the semantic attributes `realms of activity'
(government, political parties, unions and associations,
corporations) and `level' (local, regional, federal) are
represented by color and shape, respectively.

Even without any background knowledge, it is readily observed
that fairly coordinated high-level governmental actors (blue
rhomboids) dominate the structure in both relations and in both
sectors. At the top of the decisive power structure
(consideration of interest) in both cases is the board of
directors of the privatization agency (THA). This suggests that
the formal institutional framework had a great impact in the
formation of the decision system, although the positions in the
structures of `mandatory report' do not transform directly into
the final power structure based on `consideration of interest'.

It seems that especially the administrative actors (blue
rhomboids) derive a lot of their power from formal decision
rights and competencies, whereas the parties in the state
parliaments (RP, purple rectangles) cannot capitalize on their
formal rights (a rather low status value in the final power
structure), a phenomenon that has been well known and discussed
in political science for some time. This knowledge, easily gained
from looking at the visualized graphs, could then be used to
conduct further analyses about the relationship between the
structures based on different types of ties for example by using
the QAP procedure.

One can further see that the final power structure was
dominated in both cases by actors from the privatization agency
(THA), the federal government (FG), the state government (SG) and
the European Commission (EC) which very much determined the
outcome of both privatization processes. Especially in the
decision process of the privatization of the steel plants the
administrative actors' interests prevailed against the interests
of the West German industrial actors (pink ellipses on the
bottom), who called for the closure or further downsizing of the
plants.

The hierarchization of the final power structure
(`consideration of interest'), and therefore the power
concentration, is 53% for shipbuilding and 63% for steel. The
hierarchization of the structure based on `obligation of report'
for both cases is almost complete (99%). Therefore the formal
institutional structure can be regarded as highly hierarchical.
Looking more closely at the visualizations of the final power
structure (`consideration of interest'), it can be seen that the
executive governmental actors (blue rhomboids) formed a powerful
coalition and a decision core because they to a large extent
mutually considered their interests (green lines). The
hierarchization between only these actors is 0%. The overall
hierarchization is therefore created by private and societal
actors considering the interests of the political-administrative
ones (black lines bottom to top). In contrast, there are only a
few red lines (top to bottom) mainly to the European Commission,
the union (WU) and the actors within the East German enterprises
(EGE). This gives some indication that the latter two were
coopted in the decision process which was confirmed by looking at
the overall quantitative and qualitative data.

One can therefore state from the visualizations and the
hierarchization indices that a decision system with a medium
concentrated power structure developed with the privatization
agency remaining the most prominent focal actor, which could
ultimately keep control over the processes backed up by the
political and administrative actors in the federal and state
governments which formed a powerful decision core.

We have presented an approach for status visualization of
network actors. It has been shown how connecting substantive
research questions with appropriate design principles and the
appropriate algorithms can form a powerful analytic tool for the
exploration of social structures. Its greatest advantage clearly
lies in the joint representation of raw and aggregate data. Both,
links and status positions are visible and can be analyzed
together. The visualizations not only improve the communication
of results but built on the early practice of using
visualizations as an exploratory tool in structural analysis.

In the study presented in the previous section, it would have
been much harder if not impossible to detect the decision core
and the different zones of hierarchization within the decision
system and to make statements about the factors determining the
overall level of hierarchization.

A major potential of the presented approach lies in the
availability of a macroperspective, because the combined data
reveal properties of the whole or parts of the structure and make
comparison of network structures much easier:

The overall hierarchical structure of a system is
intuitively visible (positioning and distribution of
actors).

The relative status of different types or groups of
actors can be captured quickly. In the analysis of
political decision making, coalitions and decision cores
can be detected.

Analysis of what or who causes a system to be
hierarchical is possible. Relations and `zones' of
non-hierarchy are detected easily. By combining
attributes, status scores and relations, brokers and
bridges can be detected more readily.

The comparison between different social systems is much
easier. In the two cases presented here it can be easily
seen that the structures are very similar in major
properties.

Nevertheless, links and position of each actor can also be
analyzed in detail. It is possible to analyze which choices cause
the position of an actor. Is it only one choice by an actor with
a high status score that determines the position or are several
choices from actors with lower scores the reason? Is the one
actor totally dependent on only one other actor or is her ``power
base'' broader? Therefore, in closely analyzing the edges on the
micro level the stability of the actors' positions can be
determined. Compared to an analysis without visualization, it is
also easier to detect possible data entry errors. Because the
structure of choices is instantly visible, the researcher has
quick access to both raw and aggregate data and can easily check
the accuracy, if she has doubts about the position of an actor.

Though our visualizations proved useful in several
applications, we feel that a number of details - particularly in
regards to bend point placement inside clustered layers - need
further improvement. In addition, we would like to provide
automatic help for label placement, which has been refined
manually for the above examples. We need to further explore means
of user interaction: what kind of improvements may users make
without running the risk of unconsciously introducing subjective
biases?

A major line of future research will be concerned with
explanatory visualization of other types of substance, in
particular substance that, unlike centrality, status, or
clustering, does not have an immediate geometric connotation.

The visone software tool for analysis and visualization of
social networks (Baur et al., 2001)
makes the approach described in this paper available to the
general user.

Acknowledgments. The data used in Section 2 are courtesy of David
Krackhardt. Frank Müller implemented an early prototype of our
layout system in C++ using LEDA (Mehlhorn
and Näher, 1999), AGD (Mutzel
et al., 1998), and LAPACK (Anderson et al., 1999). We
thank our referees for useful comments, Rachel Lindsay for
careful proof-reading of the manuscript, and Patrick Kenis and
Volker Schneider for stimulating discussions on the subject.

Footnotes

Department of Computer & Information Science,
University of Konstanz, Ulrik.Brandes@uni-konstanz.de.
Part of this research was done while with the Department
of Computer Science at Brown University. I gratefully
acknowledge the German Academic Exchange Service
(DAAD/Hochschulsonderprogramm III) for financial
support.

Department of Public Policy & Management, University
of Konstanz, Joerg.Raab@uni-konstanz.de. Part of
this research was done while with the School of Public
Administration and Policy at the University of Arizona. I
gratefully acknowledge the Volkswagen Foundation
for financial support.

The links in the first type of tie were constructed on
the basis of the formal rules within the German
constitution, the laws in the industrial sectors and
those relevant for German unification and the
transformation of the East German economy as well as
formal agreements between the actors to establish
information and consultation rights. The links in the
latter were determined by asking representatives of the
organizations to name up to six other actors whose
interests, goals, decisions, or expectations where taken
into account in the decision making of their own
bodies/organizations.